Expander codes over reals, Euclidean sections, and compressed sensing

Classical results from the 1970´s state that w.h.p. a random subspace of N-dimensional Euclidean space of proportional (linear in N) dimension is ¿well-spread¿ in the sense that vectors in the subspace have their f₂ mass smoothly spread over a linear number of coordinates. Such well-spread subspaces are intimately connected to low distortion embeddings, compressed sensing matrices, and error-correction over reals. We describe a construction inspired by expander/Tanner codes that can be used to produce well-spread subspaces of ¿(N) dimension using sub-linear randomness (or in sub-exponential time). These results were presented in our paper. We also discuss the connection of our subspaces to compressed sensing, and describe a near-linear time iterative recovery algorithm for compressible signals.